Behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$

Question

I want to analyze the behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$, i.e I want to show it behaves like $e^x(\frac{2}{x}+\frac{2}{3x^2}+...)$ as $x\rightarrow \infty$

I started by looking the at the exponent $\cos\sqrt{t}$, taking the derivative and setting it $0$ gives me $0,\pi^2,2^2\pi^2,3^2\pi^2,...$ My guess is to use Lapalce Method now but I do not know how.

How can I continue here or is there a simpler trick for this specific integral?

Answer 1

The function $f(t) = \cos(\sqrt{t})$ has a maximum at $t = 0$ on the interval $[0,\pi^2/4]$, and near $t = 0$ we have

$$ \cos(\sqrt{t}) = 1 - \frac{1}{2}t + O(t^2), $$

so this suggests the change of variables

$$ \cos(\sqrt{t}) = 1 - \frac{1}{2}u. $$

Following this change, apply Watson's lemma.